LMFDB - Embedded newform 5796.2.p.a.3725.38

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5796,2,Mod(3725,5796)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5796, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("5796.3725");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5796 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5796.p (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$46.2812930115$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3725.38
Character	$\chi$	$=$	5796.3725
Dual form			5796.2.p.a.3725.37

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.52995 q^{5} +1.00000i q^{7} +O(q^{10})$	$q+2.52995 q^{5} +1.00000i q^{7} +0.148938 q^{11} -4.12072 q^{13} -4.48046 q^{17} -0.775942i q^{19} +(-3.94061 + 2.73342i) q^{23} +1.40063 q^{25} -4.12003i q^{29} +4.31955 q^{31} +2.52995i q^{35} -4.62073i q^{37} -1.74012i q^{41} +1.88882i q^{43} -6.00777i q^{47} -1.00000 q^{49} -1.22127 q^{53} +0.376804 q^{55} -12.6143i q^{59} +0.327176i q^{61} -10.4252 q^{65} -8.77070i q^{67} -9.70104i q^{71} -5.80529 q^{73} +0.148938i q^{77} -11.3457i q^{79} -0.932507 q^{83} -11.3353 q^{85} +7.18541 q^{89} -4.12072i q^{91} -1.96309i q^{95} -5.77969i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q+O(q^{10}) $ 48 * q	$ 48 q + 16 q^{13} + 48 q^{25} - 32 q^{31} - 48 q^{49} - 16 q^{55} - 16 q^{73} - 80 q^{85}+O(q^{100}) $ 48 * q + 16 * q^13 + 48 * q^25 - 32 * q^31 - 48 * q^49 - 16 * q^55 - 16 * q^73 - 80 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5796\mathbb{Z}\right)^\times$.

$n$	$829$	$1289$	$2899$	$4789$
$\chi(n)$	$1$	$-1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	2.52995			1.13143			0.565713	−	0.824602i	$-0.308601\pi$
$5$	2.52995			1.13143			0.565713	+	0.824602i	$0.308601\pi$
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.148938			0.0449064			0.0224532	−	0.999748i	$-0.492852\pi$
$11$	0.148938			0.0449064			0.0224532	+	0.999748i	$0.492852\pi$
$12$		0			0
$13$	−4.12072			−1.14288			−0.571441	−	0.820643i	$-0.693616\pi$
$13$	−4.12072			−1.14288			−0.571441	+	0.820643i	$0.693616\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4.48046			−1.08667			−0.543336	−	0.839516i	$-0.682839\pi$
$17$	−4.48046			−1.08667			−0.543336	+	0.839516i	$0.682839\pi$
$18$		0			0
$19$		−	0.775942i		−	0.178013i	−0.996031	−	0.0890067i	$-0.971631\pi$
$19$		−	0.775942i		−	0.178013i	0.996031	−	0.0890067i	$-0.0283692\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.94061	+	2.73342i	−0.821675	+	0.569957i
$23$	−3.94061	+	2.73342i	−0.821675	+	0.569957i
$24$		0			0
$25$	1.40063			0.280126
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	4.12003i		−	0.765070i	−0.923941	−	0.382535i	$-0.875051\pi$
$29$		−	4.12003i		−	0.765070i	0.923941	−	0.382535i	$-0.124949\pi$
$30$		0			0
$31$	4.31955			0.775814			0.387907	−	0.921698i	$-0.373198\pi$
$31$	4.31955			0.775814			0.387907	+	0.921698i	$0.373198\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			2.52995i			0.427639i
$36$		0			0
$37$		−	4.62073i		−	0.759643i	−0.925060	−	0.379822i	$-0.875985\pi$
$37$		−	4.62073i		−	0.759643i	0.925060	−	0.379822i	$-0.124015\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	1.74012i		−	0.271761i	−0.990725	−	0.135881i	$-0.956614\pi$
$41$		−	1.74012i		−	0.271761i	0.990725	−	0.135881i	$-0.0433863\pi$
$42$		0			0
$43$			1.88882i			0.288042i	0.989575	+	0.144021i	$0.0460034\pi$
$43$			1.88882i			0.288042i	−0.989575	+	0.144021i	$0.953997\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	6.00777i		−	0.876323i	−0.898896	−	0.438162i	$-0.855630\pi$
$47$		−	6.00777i		−	0.876323i	0.898896	−	0.438162i	$-0.144370\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−1.22127			−0.167755			−0.0838774	−	0.996476i	$-0.526730\pi$
$53$	−1.22127			−0.167755			−0.0838774	+	0.996476i	$0.526730\pi$
$54$		0			0
$55$	0.376804			0.0508083
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	12.6143i		−	1.64225i	−0.570750	−	0.821124i	$-0.693347\pi$
$59$		−	12.6143i		−	1.64225i	0.570750	−	0.821124i	$-0.306653\pi$
$60$		0			0
$61$			0.327176i			0.0418906i	0.999781	+	0.0209453i	$0.00666758\pi$
$61$			0.327176i			0.0418906i	−0.999781	+	0.0209453i	$0.993332\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−10.4252			−1.29309
$66$		0			0
$67$		−	8.77070i		−	1.07151i	−0.844373	−	0.535755i	$-0.820027\pi$
$67$		−	8.77070i		−	1.07151i	0.844373	−	0.535755i	$-0.179973\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	9.70104i		−	1.15130i	−0.817695	−	0.575651i	$-0.804749\pi$
$71$		−	9.70104i		−	1.15130i	0.817695	−	0.575651i	$-0.195251\pi$
$72$		0			0
$73$	−5.80529			−0.679458			−0.339729	−	0.940523i	$-0.610335\pi$
$73$	−5.80529			−0.679458			−0.339729	+	0.940523i	$0.610335\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			0.148938i			0.0169730i
$78$		0			0
$79$		−	11.3457i		−	1.27649i	−0.769835	−	0.638243i	$-0.779661\pi$
$79$		−	11.3457i		−	1.27649i	0.769835	−	0.638243i	$-0.220339\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−0.932507			−0.102356			−0.0511779	−	0.998690i	$-0.516298\pi$
$83$	−0.932507			−0.102356			−0.0511779	+	0.998690i	$0.516298\pi$
$84$		0			0
$85$	−11.3353			−1.22949
$86$		0			0
$87$		0			0
$88$		0			0
$89$	7.18541			0.761652			0.380826	−	0.924647i	$-0.375640\pi$
$89$	7.18541			0.761652			0.380826	+	0.924647i	$0.375640\pi$
$90$		0			0
$91$		−	4.12072i		−	0.431969i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	1.96309i		−	0.201409i
$96$		0			0
$97$		−	5.77969i		−	0.586838i	−0.955984	−	0.293419i	$-0.905207\pi$
$97$		−	5.77969i		−	0.586838i	0.955984	−	0.293419i	$-0.0947931\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	4.61101i		−	0.458813i	−0.973331	−	0.229406i	$-0.926321\pi$
$101$		−	4.61101i		−	0.458813i	0.973331	−	0.229406i	$-0.0736785\pi$
$102$		0			0
$103$		−	5.67463i		−	0.559138i	−0.960126	−	0.279569i	$-0.909808\pi$
$103$		−	5.67463i		−	0.559138i	0.960126	−	0.279569i	$-0.0901916\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	3.71675			0.359312			0.179656	−	0.983730i	$-0.442502\pi$
$107$	3.71675			0.359312			0.179656	+	0.983730i	$0.442502\pi$
$108$		0			0
$109$		−	6.85947i		−	0.657018i	−0.944501	−	0.328509i	$-0.893454\pi$
$109$		−	6.85947i		−	0.657018i	0.944501	−	0.328509i	$-0.106546\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	7.84409			0.737910			0.368955	−	0.929447i	$-0.379716\pi$
$113$	7.84409			0.737910			0.368955	+	0.929447i	$0.379716\pi$
$114$		0			0
$115$	−9.96954	+	6.91540i	−0.929664	+	0.644864i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	4.48046i		−	0.410723i
$120$		0			0
$121$	−10.9778			−0.997983
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−9.10622			−0.814485
$126$		0			0
$127$	7.23880			0.642340			0.321170	−	0.947022i	$-0.395924\pi$
$127$	7.23880			0.642340			0.321170	+	0.947022i	$0.395924\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			6.53804i			0.571231i	0.958344	+	0.285616i	$0.0921981\pi$
$131$			6.53804i			0.571231i	−0.958344	+	0.285616i	$0.907802\pi$
$132$		0			0
$133$	0.775942			0.0672827
$134$		0			0
$135$		0			0
$136$		0			0
$137$	19.8768			1.69819			0.849097	−	0.528237i	$-0.177147\pi$
$137$	19.8768			1.69819			0.849097	+	0.528237i	$0.177147\pi$
$138$		0			0
$139$	−17.6989			−1.50120			−0.750602	−	0.660755i	$-0.770236\pi$
$139$	−17.6989			−1.50120			−0.750602	+	0.660755i	$0.770236\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−0.613731			−0.0513227
$144$		0			0
$145$		−	10.4235i		−	0.865621i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	20.9157			1.71348			0.856739	−	0.515750i	$-0.172487\pi$
$149$	20.9157			1.71348			0.856739	+	0.515750i	$0.172487\pi$
$150$		0			0
$151$	5.51755			0.449012			0.224506	−	0.974473i	$-0.427923\pi$
$151$	5.51755			0.449012			0.224506	+	0.974473i	$0.427923\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	10.9282			0.877777
$156$		0			0
$157$			23.2541i			1.85588i	0.372731	+	0.927939i	$0.378421\pi$
$157$			23.2541i			1.85588i	−0.372731	+	0.927939i	$0.621579\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−2.73342	−	3.94061i	−0.215423	−	0.310564i
$162$		0			0
$163$	−23.1465			−1.81297			−0.906487	−	0.422233i	$-0.861246\pi$
$163$	−23.1465			−1.81297			−0.906487	+	0.422233i	$0.861246\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			11.9199i			0.922393i	0.887298	+	0.461196i	$0.152580\pi$
$167$			11.9199i			0.922393i	−0.887298	+	0.461196i	$0.847420\pi$
$168$		0			0
$169$	3.98035			0.306181
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	4.55899i		−	0.346613i	−0.984868	−	0.173307i	$-0.944555\pi$
$173$		−	4.55899i		−	0.346613i	0.984868	−	0.173307i	$-0.0554452\pi$
$174$		0			0
$175$			1.40063i			0.105878i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	4.80276i		−	0.358975i	−0.983760	−	0.179488i	$-0.942556\pi$
$179$		−	4.80276i		−	0.358975i	0.983760	−	0.179488i	$-0.0574440\pi$
$180$		0			0
$181$		−	3.32081i		−	0.246834i	−0.992355	−	0.123417i	$-0.960615\pi$
$181$		−	3.32081i		−	0.246834i	0.992355	−	0.123417i	$-0.0393852\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	11.6902i		−	0.859481i
$186$		0			0
$187$	−0.667309			−0.0487985
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−19.9748			−1.44533			−0.722663	−	0.691201i	$-0.757082\pi$
$191$	−19.9748			−1.44533			−0.722663	+	0.691201i	$0.757082\pi$
$192$		0			0
$193$	11.6553			0.838969			0.419484	−	0.907763i	$-0.362211\pi$
$193$	11.6553			0.838969			0.419484	+	0.907763i	$0.362211\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			21.4947i			1.53144i	0.643176	+	0.765718i	$0.277616\pi$
$197$			21.4947i			1.53144i	−0.643176	+	0.765718i	$0.722384\pi$
$198$		0			0
$199$		−	17.2960i		−	1.22608i	−0.790052	−	0.613040i	$-0.789947\pi$
$199$		−	17.2960i		−	1.22608i	0.790052	−	0.613040i	$-0.210053\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	4.12003			0.289169
$204$		0			0
$205$		−	4.40241i		−	0.307478i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	0.115567i		−	0.00799394i
$210$		0			0
$211$	−10.0305			−0.690530			−0.345265	−	0.938505i	$-0.612211\pi$
$211$	−10.0305			−0.690530			−0.345265	+	0.938505i	$0.612211\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			4.77862i			0.325899i
$216$		0			0
$217$			4.31955i			0.293230i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	18.4627			1.24194
$222$		0			0
$223$	−6.09800			−0.408352			−0.204176	−	0.978934i	$-0.565451\pi$
$223$	−6.09800			−0.408352			−0.204176	+	0.978934i	$0.565451\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	20.5885			1.36651			0.683254	−	0.730181i	$-0.260564\pi$
$227$	20.5885			1.36651			0.683254	+	0.730181i	$0.260564\pi$
$228$		0			0
$229$			16.0052i			1.05765i	0.848730	+	0.528827i	$0.177368\pi$
$229$			16.0052i			1.05765i	−0.848730	+	0.528827i	$0.822632\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			24.7086i			1.61871i	0.587318	+	0.809356i	$0.300184\pi$
$233$			24.7086i			1.61871i	−0.587318	+	0.809356i	$0.699816\pi$
$234$		0			0
$235$		−	15.1993i		−	0.991495i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	14.2201i		−	0.919822i	−0.887965	−	0.459911i	$-0.847881\pi$
$239$		−	14.2201i		−	0.919822i	0.887965	−	0.459911i	$-0.152119\pi$
$240$		0			0
$241$			6.99000i			0.450265i	0.974328	+	0.225133i	$0.0722815\pi$
$241$			6.99000i			0.450265i	−0.974328	+	0.225133i	$0.927718\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2.52995			−0.161632
$246$		0			0
$247$			3.19744i			0.203448i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	7.51447			0.474309			0.237155	−	0.971472i	$-0.423785\pi$
$251$	7.51447			0.474309			0.237155	+	0.971472i	$0.423785\pi$
$252$		0			0
$253$	−0.586906	+	0.407109i	−0.0368984	+	0.0255947i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	22.4100i		−	1.39790i	−0.715172	−	0.698948i	$-0.753652\pi$
$257$		−	22.4100i		−	1.39790i	0.715172	−	0.698948i	$-0.246348\pi$
$258$		0			0
$259$	4.62073			0.287118
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−1.87715			−0.115750			−0.0578750	−	0.998324i	$-0.518432\pi$
$263$	−1.87715			−0.115750			−0.0578750	+	0.998324i	$0.518432\pi$
$264$		0			0
$265$	−3.08976			−0.189802
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	6.99760i		−	0.426651i	−0.976981	−	0.213326i	$-0.931570\pi$
$269$		−	6.99760i		−	0.426651i	0.976981	−	0.213326i	$-0.0684295\pi$
$270$		0			0
$271$	−5.82619			−0.353916			−0.176958	−	0.984218i	$-0.556626\pi$
$271$	−5.82619			−0.353916			−0.176958	+	0.984218i	$0.556626\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.208606			0.0125794
$276$		0			0
$277$	−23.3172			−1.40100			−0.700498	−	0.713654i	$-0.747039\pi$
$277$	−23.3172			−1.40100			−0.700498	+	0.713654i	$0.747039\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−2.02569			−0.120843			−0.0604214	−	0.998173i	$-0.519244\pi$
$281$	−2.02569			−0.120843			−0.0604214	+	0.998173i	$0.519244\pi$
$282$		0			0
$283$		−	11.0762i		−	0.658410i	−0.944258	−	0.329205i	$-0.893219\pi$
$283$		−	11.0762i		−	0.658410i	0.944258	−	0.329205i	$-0.106781\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	1.74012			0.102716
$288$		0			0
$289$	3.07453			0.180855
$290$		0			0
$291$		0			0
$292$		0			0
$293$	0.0960819			0.00561316			0.00280658	−	0.999996i	$-0.499107\pi$
$293$	0.0960819			0.00561316			0.00280658	+	0.999996i	$0.499107\pi$
$294$		0			0
$295$		−	31.9136i		−	1.85808i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	16.2382	−	11.2637i	0.939078	−	0.651394i
$300$		0			0
$301$	−1.88882			−0.108870
$302$		0			0
$303$		0			0
$304$		0			0
$305$			0.827737i			0.0473961i
$306$		0			0
$307$	7.91870			0.451944			0.225972	−	0.974134i	$-0.427444\pi$
$307$	7.91870			0.451944			0.225972	+	0.974134i	$0.427444\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	16.8699i		−	0.956603i	−0.878196	−	0.478301i	$-0.841253\pi$
$311$		−	16.8699i		−	0.956603i	0.878196	−	0.478301i	$-0.158747\pi$
$312$		0			0
$313$			25.1608i			1.42217i	0.703106	+	0.711085i	$0.251796\pi$
$313$			25.1608i			1.42217i	−0.703106	+	0.711085i	$0.748204\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			6.20949i			0.348760i	0.984678	+	0.174380i	$0.0557921\pi$
$317$			6.20949i			0.348760i	−0.984678	+	0.174380i	$0.944208\pi$
$318$		0			0
$319$		−	0.613628i		−	0.0343566i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			3.47658i			0.193442i
$324$		0			0
$325$	−5.77160			−0.320151
$326$		0			0
$327$		0			0
$328$		0			0
$329$	6.00777			0.331219
$330$		0			0
$331$	−35.5936			−1.95640			−0.978201	−	0.207659i	$-0.933416\pi$
$331$	−35.5936			−1.95640			−0.978201	+	0.207659i	$0.933416\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	22.1894i		−	1.21234i
$336$		0			0
$337$			0.673930i			0.0367113i	0.999832	+	0.0183557i	$0.00584312\pi$
$337$			0.673930i			0.0367113i	−0.999832	+	0.0183557i	$0.994157\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	0.643344			0.0348390
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	8.80190i		−	0.472511i	−0.971691	−	0.236255i	$-0.924080\pi$
$347$		−	8.80190i		−	0.472511i	0.971691	−	0.236255i	$-0.0759202\pi$
$348$		0			0
$349$	−28.4731			−1.52413			−0.762066	−	0.647499i	$-0.775815\pi$
$349$	−28.4731			−1.52413			−0.762066	+	0.647499i	$0.775815\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			14.8208i			0.788832i	0.918932	+	0.394416i	$0.129053\pi$
$353$			14.8208i			0.788832i	−0.918932	+	0.394416i	$0.870947\pi$
$354$		0			0
$355$		−	24.5431i		−	1.30261i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−12.4115			−0.655054			−0.327527	−	0.944842i	$-0.606215\pi$
$359$	−12.4115			−0.655054			−0.327527	+	0.944842i	$0.606215\pi$
$360$		0			0
$361$	18.3979			0.968311
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−14.6871			−0.768757
$366$		0			0
$367$			34.9854i			1.82622i	0.407713	+	0.913110i	$0.366327\pi$
$367$			34.9854i			1.82622i	−0.407713	+	0.913110i	$0.633673\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	1.22127i		−	0.0634053i
$372$		0			0
$373$			7.84130i			0.406007i	0.979178	+	0.203004i	$0.0650703\pi$
$373$			7.84130i			0.406007i	−0.979178	+	0.203004i	$0.934930\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			16.9775i			0.874386i
$378$		0			0
$379$		−	9.86934i		−	0.506954i	−0.967341	−	0.253477i	$-0.918426\pi$
$379$		−	9.86934i		−	0.506954i	0.967341	−	0.253477i	$-0.0815742\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	2.57892			0.131776			0.0658882	−	0.997827i	$-0.479012\pi$
$383$	2.57892			0.131776			0.0658882	+	0.997827i	$0.479012\pi$
$384$		0			0
$385$			0.376804i			0.0192037i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	3.73943			0.189597			0.0947984	−	0.995496i	$-0.469779\pi$
$389$	3.73943			0.189597			0.0947984	+	0.995496i	$0.469779\pi$
$390$		0			0
$391$	17.6558	−	12.2470i	0.892890	−	0.619356i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	28.7039i		−	1.44425i
$396$		0			0
$397$	1.50547			0.0755571			0.0377786	−	0.999286i	$-0.487972\pi$
$397$	1.50547			0.0755571			0.0377786	+	0.999286i	$0.487972\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−1.42987			−0.0714043			−0.0357021	−	0.999362i	$-0.511367\pi$
$401$	−1.42987			−0.0714043			−0.0357021	+	0.999362i	$0.511367\pi$
$402$		0			0
$403$	−17.7997			−0.886665
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	0.688201i		−	0.0341128i
$408$		0			0
$409$	−36.1330			−1.78666			−0.893330	−	0.449401i	$-0.851637\pi$
$409$	−36.1330			−1.78666			−0.893330	+	0.449401i	$0.851637\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	12.6143			0.620712
$414$		0			0
$415$	−2.35919			−0.115808
$416$		0			0
$417$		0			0
$418$		0			0
$419$	13.8636			0.677282			0.338641	−	0.940916i	$-0.390033\pi$
$419$	13.8636			0.677282			0.338641	+	0.940916i	$0.390033\pi$
$420$		0			0
$421$		−	24.0942i		−	1.17428i	−0.809485	−	0.587140i	$-0.800254\pi$
$421$		−	24.0942i		−	1.17428i	0.809485	−	0.587140i	$-0.199746\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−6.27546			−0.304405
$426$		0			0
$427$	−0.327176			−0.0158331
$428$		0			0
$429$		0			0
$430$		0			0
$431$	25.8172			1.24357			0.621786	−	0.783187i	$-0.286407\pi$
$431$	25.8172			1.24357			0.621786	+	0.783187i	$0.286407\pi$
$432$		0			0
$433$		−	28.7807i		−	1.38311i	−0.722323	−	0.691555i	$-0.756926\pi$
$433$		−	28.7807i		−	1.38311i	0.722323	−	0.691555i	$-0.243074\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	2.12097	+	3.05769i	0.101460	+	0.146269i
$438$		0			0
$439$	−23.5643			−1.12466			−0.562332	−	0.826911i	$-0.690096\pi$
$439$	−23.5643			−1.12466			−0.562332	+	0.826911i	$0.690096\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	20.6291i		−	0.980120i	−0.871689	−	0.490060i	$-0.836975\pi$
$443$		−	20.6291i		−	0.980120i	0.871689	−	0.490060i	$-0.163025\pi$
$444$		0			0
$445$	18.1787			0.861754
$446$		0			0
$447$		0			0
$448$		0			0
$449$			19.6011i			0.925032i	0.886611	+	0.462516i	$0.153053\pi$
$449$			19.6011i			0.925032i	−0.886611	+	0.462516i	$0.846947\pi$
$450$		0			0
$451$		−	0.259169i		−	0.0122038i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	10.4252i		−	0.488741i
$456$		0			0
$457$			4.70323i			0.220008i	0.993931	+	0.110004i	$0.0350864\pi$
$457$			4.70323i			0.220008i	−0.993931	+	0.110004i	$0.964914\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			19.9401i			0.928703i	0.885651	+	0.464351i	$0.153713\pi$
$461$			19.9401i			0.928703i	−0.885651	+	0.464351i	$0.846287\pi$
$462$		0			0
$463$	3.07647			0.142976			0.0714879	−	0.997441i	$-0.477225\pi$
$463$	3.07647			0.142976			0.0714879	+	0.997441i	$0.477225\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−11.8091			−0.546460			−0.273230	−	0.961949i	$-0.588092\pi$
$467$	−11.8091			−0.546460			−0.273230	+	0.961949i	$0.588092\pi$
$468$		0			0
$469$	8.77070			0.404993
$470$		0			0
$471$		0			0
$472$		0			0
$473$			0.281317i			0.0129349i
$474$		0			0
$475$		−	1.08681i		−	0.0498661i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−21.0311			−0.960934			−0.480467	−	0.877013i	$-0.659533\pi$
$479$	−21.0311			−0.960934			−0.480467	+	0.877013i	$0.659533\pi$
$480$		0			0
$481$			19.0407i			0.868183i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	14.6223i		−	0.663964i
$486$		0			0
$487$	−8.20875			−0.371974			−0.185987	−	0.982552i	$-0.559548\pi$
$487$	−8.20875			−0.371974			−0.185987	+	0.982552i	$0.559548\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			9.58335i			0.432491i	0.976339	+	0.216245i	$0.0693811\pi$
$491$			9.58335i			0.432491i	−0.976339	+	0.216245i	$0.930619\pi$
$492$		0			0
$493$			18.4596i			0.831380i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	9.70104			0.435151
$498$		0			0
$499$	−31.4415			−1.40751			−0.703757	−	0.710441i	$-0.748496\pi$
$499$	−31.4415			−1.40751			−0.703757	+	0.710441i	$0.748496\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−1.21401			−0.0541301			−0.0270650	−	0.999634i	$-0.508616\pi$
$503$	−1.21401			−0.0541301			−0.0270650	+	0.999634i	$0.508616\pi$
$504$		0			0
$505$		−	11.6656i		−	0.519113i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	15.2568i		−	0.676246i	−0.941102	−	0.338123i	$-0.890208\pi$
$509$		−	15.2568i		−	0.676246i	0.941102	−	0.338123i	$-0.109792\pi$
$510$		0			0
$511$		−	5.80529i		−	0.256811i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	14.3565i		−	0.632624i
$516$		0			0
$517$		−	0.894783i		−	0.0393525i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−5.03937			−0.220779			−0.110389	−	0.993888i	$-0.535210\pi$
$521$	−5.03937			−0.220779			−0.110389	+	0.993888i	$0.535210\pi$
$522$		0			0
$523$		−	5.11943i		−	0.223857i	−0.993716	−	0.111929i	$-0.964297\pi$
$523$		−	5.11943i		−	0.223857i	0.993716	−	0.111929i	$-0.0357028\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−19.3536			−0.843055
$528$		0			0
$529$	8.05687	−	21.5427i	0.350299	−	0.936638i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			7.17055i			0.310591i
$534$		0			0
$535$	9.40318			0.406535
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−0.148938			−0.00641520
$540$		0			0
$541$	44.4139			1.90950			0.954752	−	0.297402i	$-0.0961203\pi$
$541$	44.4139			1.90950			0.954752	+	0.297402i	$0.0961203\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	17.3541i		−	0.743368i
$546$		0			0
$547$	−27.9106			−1.19337			−0.596686	−	0.802475i	$-0.703516\pi$
$547$	−27.9106			−1.19337			−0.596686	+	0.802475i	$0.703516\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−3.19690			−0.136193
$552$		0			0
$553$	11.3457			0.482467
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−27.6681			−1.17233			−0.586167	−	0.810190i	$-0.699364\pi$
$557$	−27.6681			−1.17233			−0.586167	+	0.810190i	$0.699364\pi$
$558$		0			0
$559$		−	7.78331i		−	0.329199i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	27.2314			1.14767			0.573833	−	0.818972i	$-0.305456\pi$
$563$	27.2314			1.14767			0.573833	+	0.818972i	$0.305456\pi$
$564$		0			0
$565$	19.8451			0.834891
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−5.12271			−0.214755			−0.107378	−	0.994218i	$-0.534245\pi$
$569$	−5.12271			−0.214755			−0.107378	+	0.994218i	$0.534245\pi$
$570$		0			0
$571$		−	1.56201i		−	0.0653680i	−0.999466	−	0.0326840i	$-0.989595\pi$
$571$		−	1.56201i		−	0.0653680i	0.999466	−	0.0326840i	$-0.0104055\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−5.51934	+	3.82850i	−0.230172	+	0.159660i
$576$		0			0
$577$	−4.94862			−0.206014			−0.103007	−	0.994681i	$-0.532846\pi$
$577$	−4.94862			−0.206014			−0.103007	+	0.994681i	$0.532846\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	0.932507i		−	0.0386869i
$582$		0			0
$583$	−0.181894			−0.00753326
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	8.23518i		−	0.339902i	−0.985452	−	0.169951i	$-0.945639\pi$
$587$		−	8.23518i		−	0.339902i	0.985452	−	0.169951i	$-0.0543610\pi$
$588$		0			0
$589$		−	3.35172i		−	0.138105i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	35.8156i		−	1.47077i	−0.677649	−	0.735386i	$-0.737001\pi$
$593$		−	35.8156i		−	1.47077i	0.677649	−	0.735386i	$-0.262999\pi$
$594$		0			0
$595$		−	11.3353i		−	0.464703i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	22.8844i		−	0.935031i	−0.883985	−	0.467515i	$-0.845149\pi$
$599$		−	22.8844i		−	0.935031i	0.883985	−	0.467515i	$-0.154851\pi$
$600$		0			0
$601$	34.7863			1.41896			0.709481	−	0.704725i	$-0.248930\pi$
$601$	34.7863			1.41896			0.709481	+	0.704725i	$0.248930\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−27.7733			−1.12914
$606$		0			0
$607$	10.7198			0.435104			0.217552	−	0.976049i	$-0.430193\pi$
$607$	10.7198			0.435104			0.217552	+	0.976049i	$0.430193\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			24.7563i			1.00153i
$612$		0			0
$613$			29.5254i			1.19252i	0.802792	+	0.596260i	$0.203347\pi$
$613$			29.5254i			1.19252i	−0.802792	+	0.596260i	$0.796653\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	9.60152			0.386543			0.193271	−	0.981145i	$-0.438090\pi$
$617$	9.60152			0.386543			0.193271	+	0.981145i	$0.438090\pi$
$618$		0			0
$619$			26.5651i			1.06774i	0.845566	+	0.533871i	$0.179263\pi$
$619$			26.5651i			1.06774i	−0.845566	+	0.533871i	$0.820737\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			7.18541i			0.287878i
$624$		0			0
$625$	−30.0414			−1.20166
$626$		0			0
$627$		0			0
$628$		0			0
$629$			20.7030i			0.825483i
$630$		0			0
$631$		−	19.8921i		−	0.791893i	−0.918274	−	0.395946i	$-0.870417\pi$
$631$		−	19.8921i		−	0.791893i	0.918274	−	0.395946i	$-0.129583\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	18.3138			0.726760
$636$		0			0
$637$	4.12072			0.163269
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−13.6329			−0.538469			−0.269234	−	0.963075i	$-0.586771\pi$
$641$	−13.6329			−0.538469			−0.269234	+	0.963075i	$0.586771\pi$
$642$		0			0
$643$		−	6.55297i		−	0.258424i	−0.991617	−	0.129212i	$-0.958755\pi$
$643$		−	6.55297i		−	0.258424i	0.991617	−	0.129212i	$-0.0412448\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	15.4159i		−	0.606062i	−0.952981	−	0.303031i	$-0.902002\pi$
$647$		−	15.4159i		−	0.606062i	0.952981	−	0.303031i	$-0.0979985\pi$
$648$		0			0
$649$		−	1.87875i		−	0.0737474i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			4.40349i			0.172322i	0.996281	+	0.0861610i	$0.0274599\pi$
$653$			4.40349i			0.172322i	−0.996281	+	0.0861610i	$0.972540\pi$
$654$		0			0
$655$			16.5409i			0.646306i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	15.3289			0.597131			0.298565	−	0.954389i	$-0.403492\pi$
$659$	15.3289			0.597131			0.298565	+	0.954389i	$0.403492\pi$
$660$		0			0
$661$		−	5.89302i		−	0.229212i	−0.993411	−	0.114606i	$-0.963439\pi$
$661$		−	5.89302i		−	0.229212i	0.993411	−	0.114606i	$-0.0365605\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	1.96309			0.0761254
$666$		0			0
$667$	11.2618	+	16.2354i	0.436057	+	0.628639i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			0.0487288i			0.00188115i
$672$		0			0
$673$	19.6865			0.758859			0.379430	−	0.925221i	$-0.376120\pi$
$673$	19.6865			0.758859			0.379430	+	0.925221i	$0.376120\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−31.1163			−1.19590			−0.597948	−	0.801535i	$-0.704017\pi$
$677$	−31.1163			−1.19590			−0.597948	+	0.801535i	$0.704017\pi$
$678$		0			0
$679$	5.77969			0.221804
$680$		0			0
$681$		0			0
$682$		0			0
$683$			9.57583i			0.366409i	0.983075	+	0.183204i	$0.0586470\pi$
$683$			9.57583i			0.366409i	−0.983075	+	0.183204i	$0.941353\pi$
$684$		0			0
$685$	50.2874			1.92138
$686$		0			0
$687$		0			0
$688$		0			0
$689$	5.03253			0.191724
$690$		0			0
$691$	−25.2991			−0.962424			−0.481212	−	0.876604i	$-0.659803\pi$
$691$	−25.2991			−0.962424			−0.481212	+	0.876604i	$0.659803\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−44.7773			−1.69850
$696$		0			0
$697$			7.79654i			0.295315i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−23.0197			−0.869443			−0.434722	−	0.900565i	$-0.643153\pi$
$701$	−23.0197			−0.869443			−0.434722	+	0.900565i	$0.643153\pi$
$702$		0			0
$703$	−3.58542			−0.135227
$704$		0			0
$705$		0			0
$706$		0			0
$707$	4.61101			0.173415
$708$		0			0
$709$		−	2.68502i		−	0.100838i	−0.998728	−	0.0504191i	$-0.983944\pi$
$709$		−	2.68502i		−	0.100838i	0.998728	−	0.0504191i	$-0.0160557\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−17.0217	+	11.8071i	−0.637467	+	0.442181i
$714$		0			0
$715$	−1.55271			−0.0580679
$716$		0			0
$717$		0			0
$718$		0			0
$719$			3.08498i			0.115050i	0.998344	+	0.0575252i	$0.0183210\pi$
$719$			3.08498i			0.115050i	−0.998344	+	0.0575252i	$0.981679\pi$
$720$		0			0
$721$	5.67463			0.211334
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	5.77064i		−	0.214316i
$726$		0			0
$727$		−	9.05394i		−	0.335792i	−0.985805	−	0.167896i	$-0.946303\pi$
$727$		−	9.05394i		−	0.335792i	0.985805	−	0.167896i	$-0.0536973\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	8.46279i		−	0.313008i
$732$		0			0
$733$			6.63913i			0.245222i	0.992455	+	0.122611i	$0.0391267\pi$
$733$			6.63913i			0.245222i	−0.992455	+	0.122611i	$0.960873\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	1.30629i		−	0.0481177i
$738$		0			0
$739$	8.24877			0.303436			0.151718	−	0.988424i	$-0.451519\pi$
$739$	8.24877			0.303436			0.151718	+	0.988424i	$0.451519\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	7.93170			0.290986			0.145493	−	0.989359i	$-0.453523\pi$
$743$	7.93170			0.290986			0.145493	+	0.989359i	$0.453523\pi$
$744$		0			0
$745$	52.9155			1.93868
$746$		0			0
$747$		0			0
$748$		0			0
$749$			3.71675i			0.135807i
$750$		0			0
$751$		−	32.6877i		−	1.19279i	−0.802690	−	0.596396i	$-0.796599\pi$
$751$		−	32.6877i		−	1.19279i	0.802690	−	0.596396i	$-0.203401\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	13.9591			0.508024
$756$		0			0
$757$			0.710487i			0.0258231i	0.999917	+	0.0129115i	$0.00410998\pi$
$757$			0.710487i			0.0258231i	−0.999917	+	0.0129115i	$0.995890\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	8.26317i		−	0.299540i	−0.988721	−	0.149770i	$-0.952147\pi$
$761$		−	8.26317i		−	0.299540i	0.988721	−	0.149770i	$-0.0478532\pi$
$762$		0			0
$763$	6.85947			0.248330
$764$		0			0
$765$		0			0
$766$		0			0
$767$			51.9802i			1.87690i
$768$		0			0
$769$		−	20.6049i		−	0.743033i	−0.928426	−	0.371516i	$-0.878838\pi$
$769$		−	20.6049i		−	0.743033i	0.928426	−	0.371516i	$-0.121162\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−36.8271			−1.32458			−0.662290	−	0.749248i	$-0.730415\pi$
$773$	−36.8271			−1.32458			−0.662290	+	0.749248i	$0.730415\pi$
$774$		0			0
$775$	6.05009			0.217326
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−1.35023			−0.0483771
$780$		0			0
$781$		−	1.44485i		−	0.0517008i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			58.8316i			2.09979i
$786$		0			0
$787$		−	11.8097i		−	0.420972i	−0.977597	−	0.210486i	$-0.932495\pi$
$787$		−	11.8097i		−	0.420972i	0.977597	−	0.210486i	$-0.0675046\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			7.84409i			0.278904i
$792$		0			0
$793$		−	1.34820i		−	0.0478760i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	39.5755			1.40183			0.700917	−	0.713243i	$-0.252774\pi$
$797$	39.5755			1.40183			0.700917	+	0.713243i	$0.252774\pi$
$798$		0			0
$799$			26.9176i			0.952275i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−0.864626			−0.0305120
$804$		0			0
$805$	−6.91540	−	9.96954i	−0.243736	−	0.351380i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	6.43666i		−	0.226301i	−0.993578	−	0.113150i	$-0.963906\pi$
$809$		−	6.43666i		−	0.226301i	0.993578	−	0.113150i	$-0.0360942\pi$
$810$		0			0
$811$	−43.8629			−1.54023			−0.770117	−	0.637903i	$-0.779802\pi$
$811$	−43.8629			−1.54023			−0.770117	+	0.637903i	$0.779802\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−58.5594			−2.05125
$816$		0			0
$817$	1.46562			0.0512754
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	3.63643i		−	0.126912i	−0.997985	−	0.0634561i	$-0.979788\pi$
$821$		−	3.63643i		−	0.126912i	0.997985	−	0.0634561i	$-0.0202123\pi$
$822$		0			0
$823$	6.32577			0.220502			0.110251	−	0.993904i	$-0.464834\pi$
$823$	6.32577			0.220502			0.110251	+	0.993904i	$0.464834\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	37.1316			1.29119			0.645596	−	0.763679i	$-0.276609\pi$
$827$	37.1316			1.29119			0.645596	+	0.763679i	$0.276609\pi$
$828$		0			0
$829$	−33.5445			−1.16505			−0.582524	−	0.812814i	$-0.697935\pi$
$829$	−33.5445			−1.16505			−0.582524	+	0.812814i	$0.697935\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	4.48046			0.155239
$834$		0			0
$835$			30.1568i			1.04362i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−7.07428			−0.244231			−0.122116	−	0.992516i	$-0.538968\pi$
$839$	−7.07428			−0.244231			−0.122116	+	0.992516i	$0.538968\pi$
$840$		0			0
$841$	12.0253			0.414667
$842$		0			0
$843$		0			0
$844$		0			0
$845$	10.0701			0.346421
$846$		0			0
$847$		−	10.9778i		−	0.377202i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	12.6304	+	18.2085i	0.432964	+	0.624180i
$852$		0			0
$853$	−6.09610			−0.208726			−0.104363	−	0.994539i	$-0.533280\pi$
$853$	−6.09610			−0.208726			−0.104363	+	0.994539i	$0.533280\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	9.33970i		−	0.319038i	−0.987195	−	0.159519i	$-0.949006\pi$
$857$		−	9.33970i		−	0.319038i	0.987195	−	0.159519i	$-0.0509943\pi$
$858$		0			0
$859$	24.1222			0.823040			0.411520	−	0.911401i	$-0.364998\pi$
$859$	24.1222			0.823040			0.411520	+	0.911401i	$0.364998\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	16.1279i		−	0.548998i	−0.961587	−	0.274499i	$-0.911488\pi$
$863$		−	16.1279i		−	0.548998i	0.961587	−	0.274499i	$-0.0885121\pi$
$864$		0			0
$865$		−	11.5340i		−	0.392167i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	1.68980i		−	0.0573224i
$870$		0			0
$871$			36.1416i			1.22461i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	9.10622i		−	0.307846i
$876$		0			0
$877$	16.7922			0.567031			0.283516	−	0.958968i	$-0.408499\pi$
$877$	16.7922			0.567031			0.283516	+	0.958968i	$0.408499\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−8.88554			−0.299361			−0.149681	−	0.988734i	$-0.547825\pi$
$881$	−8.88554			−0.299361			−0.149681	+	0.988734i	$0.547825\pi$
$882$		0			0
$883$	35.8056			1.20496			0.602478	−	0.798136i	$-0.294180\pi$
$883$	35.8056			1.20496			0.602478	+	0.798136i	$0.294180\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			4.99655i			0.167768i	0.996476	+	0.0838839i	$0.0267325\pi$
$887$			4.99655i			0.167768i	−0.996476	+	0.0838839i	$0.973267\pi$
$888$		0			0
$889$			7.23880i			0.242782i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−4.66168			−0.155997
$894$		0			0
$895$		−	12.1507i		−	0.406154i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	17.7967i		−	0.593553i
$900$		0			0
$901$	5.47187			0.182294
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	8.40146i		−	0.279274i
$906$		0			0
$907$		−	27.7345i		−	0.920909i	−0.887683	−	0.460454i	$-0.847686\pi$
$907$		−	27.7345i		−	0.920909i	0.887683	−	0.460454i	$-0.152314\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	55.2253			1.82970			0.914848	−	0.403799i	$-0.132310\pi$
$911$	55.2253			1.82970			0.914848	+	0.403799i	$0.132310\pi$
$912$		0			0
$913$	−0.138885			−0.00459643
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−6.53804			−0.215905
$918$		0			0
$919$			23.9880i			0.791290i	0.918404	+	0.395645i	$0.129479\pi$
$919$			23.9880i			0.791290i	−0.918404	+	0.395645i	$0.870521\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			39.9753i			1.31580i
$924$		0			0
$925$		−	6.47193i		−	0.212796i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			9.74208i			0.319627i	0.987147	+	0.159814i	$0.0510893\pi$
$929$			9.74208i			0.319627i	−0.987147	+	0.159814i	$0.948911\pi$
$930$		0			0
$931$			0.775942i			0.0254305i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−1.68826			−0.0552119
$936$		0			0
$937$			49.2496i			1.60892i	0.594010	+	0.804458i	$0.297544\pi$
$937$			49.2496i			1.60892i	−0.594010	+	0.804458i	$0.702456\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−37.7486			−1.23057			−0.615284	−	0.788305i	$-0.710959\pi$
$941$	−37.7486			−1.23057			−0.615284	+	0.788305i	$0.710959\pi$
$942$		0			0
$943$	4.75647	+	6.85714i	0.154892	+	0.223299i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			38.3991i			1.24780i	0.781502	+	0.623902i	$0.214454\pi$
$947$			38.3991i			1.24780i	−0.781502	+	0.623902i	$0.785546\pi$
$948$		0			0
$949$	23.9220			0.776541
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1.57546			−0.0510342			−0.0255171	−	0.999674i	$-0.508123\pi$
$953$	−1.57546			−0.0510342			−0.0255171	+	0.999674i	$0.508123\pi$
$954$		0			0
$955$	−50.5352			−1.63528
$956$		0			0
$957$		0			0
$958$		0			0
$959$			19.8768i			0.641857i
$960$		0			0
$961$	−12.3415			−0.398112
$962$		0			0
$963$		0			0
$964$		0			0
$965$	29.4874			0.949232
$966$		0			0
$967$	19.0807			0.613594			0.306797	−	0.951775i	$-0.400743\pi$
$967$	19.0807			0.613594			0.306797	+	0.951775i	$0.400743\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	19.8466			0.636909			0.318454	−	0.947938i	$-0.396836\pi$
$971$	19.8466			0.636909			0.318454	+	0.947938i	$0.396836\pi$
$972$		0			0
$973$		−	17.6989i		−	0.567402i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−19.9970			−0.639761			−0.319880	−	0.947458i	$-0.603643\pi$
$977$	−19.9970			−0.639761			−0.319880	+	0.947458i	$0.603643\pi$
$978$		0			0
$979$	1.07018			0.0342031
$980$		0			0
$981$		0			0
$982$		0			0
$983$	34.6159			1.10408			0.552038	−	0.833819i	$-0.313850\pi$
$983$	34.6159			1.10408			0.552038	+	0.833819i	$0.313850\pi$
$984$		0			0
$985$			54.3805i			1.73271i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−5.16293	−	7.44311i	−0.164172	−	0.236677i
$990$		0			0
$991$	−29.6550			−0.942024			−0.471012	−	0.882127i	$-0.656111\pi$
$991$	−29.6550			−0.942024			−0.471012	+	0.882127i	$0.656111\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	43.7579i		−	1.38722i
$996$		0			0
$997$	−8.27099			−0.261945			−0.130972	−	0.991386i	$-0.541810\pi$
$997$	−8.27099			−0.261945			−0.130972	+	0.991386i	$0.541810\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5796.2.p.a.3725.38	yes	48
3.2	odd	2	inner	5796.2.p.a.3725.12	yes	48
23.22	odd	2	inner	5796.2.p.a.3725.11	✓	48
69.68	even	2	inner	5796.2.p.a.3725.37	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5796.2.p.a.3725.11	✓	48	23.22	odd	2	inner
5796.2.p.a.3725.12	yes	48	3.2	odd	2	inner
5796.2.p.a.3725.37	yes	48	69.68	even	2	inner
5796.2.p.a.3725.38	yes	48	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 5796 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5796.p (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.52995 q^{5} +1.00000i q^{7} +O(q^{10})\)	\(q+2.52995 q^{5} +1.00000i q^{7} +0.148938 q^{11} -4.12072 q^{13} -4.48046 q^{17} -0.775942i q^{19} +(-3.94061 + 2.73342i) q^{23} +1.40063 q^{25} -4.12003i q^{29} +4.31955 q^{31} +2.52995i q^{35} -4.62073i q^{37} -1.74012i q^{41} +1.88882i q^{43} -6.00777i q^{47} -1.00000 q^{49} -1.22127 q^{53} +0.376804 q^{55} -12.6143i q^{59} +0.327176i q^{61} -10.4252 q^{65} -8.77070i q^{67} -9.70104i q^{71} -5.80529 q^{73} +0.148938i q^{77} -11.3457i q^{79} -0.932507 q^{83} -11.3353 q^{85} +7.18541 q^{89} -4.12072i q^{91} -1.96309i q^{95} -5.77969i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 48 * q	\( 48 q + 16 q^{13} + 48 q^{25} - 32 q^{31} - 48 q^{49} - 16 q^{55} - 16 q^{73} - 80 q^{85}+O(q^{100}) \) 48 * q + 16 * q^13 + 48 * q^25 - 32 * q^31 - 48 * q^49 - 16 * q^55 - 16 * q^73 - 80 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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